Look familiar? For most people, if they ever learned to do long division properly, they had to “master” something similar to the above procedure/algorithm. What was not required was having the vaguest notion of what’s going on, why anyone should care about knowing how to do it, or why, if you did have some idea about what the answer was supposed to mean, this mumbo-jumbo got you to that answer correctly. Another way of describing the situation is “a black box.” You take two numbers, throw them into the “long division box,” turn the handle as many times as necessary, and – voila! – out pops the magic quotient, with a remainder, if any, in some form or other. Fabulous! Who needs conceptual understanding if you can memorize a procedure and carry it out accurately? Oh, and quickly: don’t forget that “math” (well, school math, anyway), is a race! The “good” students are like machines! The slow, the error-prone, the confused, will be left by the wayside.
Well, that doesn’t work for me anymore with mathematics. I can’t be satisfied with being a flesh-robot or training others to be like that (or abetting teachers to make children view math like that). Procedural fluency sans understanding is not better than using an electronic black box for computation, though traditionalists praise the former and decry the latter. There’s a deep conversation (or pointless, screaming argument complete with epithets that would make participants in a Trump vs. Clinton partisan on-line fight blush) to be had here, but not today. Instead, I’m going to draw back the curtain and show you the wizard hiding behind it.
To begin, let’s recall from last month that one way to think about computing a multiplication problem in the world of integers (Z) is as “repeated addition.” While there are problems with this phrase as a definition of multiplication, one I advise that teachers eschew, there is no problem getting the product of, say, 18 x 6 by summing six addends of eighteen or vice versa. Not ideal for efficiency, of course.
By analogy, since division is the inverse operation of multiplication and subtraction is the inverse operation of addition, we should be able to compute division in Z as repeated subtraction (as long as we are comfortable with remainders. In the rational numbers (Q), we have “fractions” or decimals when a division of integers doesn’t give a “nice” integer quotient). Here’s an example of how that would work in practice:
So, we could certainly do without long division if we’re willing to do repeated subtraction as many times as necessary to get the quotient of two integers, right? And if we are not in a hurry for results or worried that adding more than the minimum calculations necessary increases the likelihood of error, we’re fine. We really DO have that option.
But for those who prefer another approach, there’s long division. Why does it work? That’s the missing piece for most of us, including, I strongly suspect, most K-6 math teachers. I would be willing to bet that if you asked people to explain why the long-division algorithm works – not simply what the steps are, but why and how they make sense – few could give a coherent, correct response.
So, let’s try to unpack long division by applying the standard algorithm to something similar to the previous problem (we can risk a bigger dividend here):
I trust your answer agrees with mine. But how do we know we are correct? Well, 78 x 22 = 1716. Add the remainder, 14, and we get 1730, the original dividend. Math works, life is good.
Except that we don’t know why we got that result, exactly, only that we followed the recipe and almost miraculously, out popped the perfect cake.
One thing worth noting that could easily be missed: if we add 1540 + 176, we get 1716. That can’t be sheer coincidence, can it? What are those numbers, anyway?
Let’s go through the steps used to get our answer, but a bit more thoroughly than you are probably used to doing. First, you likely said, “Hmm, 22 won’t go into 17, so let’s try 22 into 173. Ah, that goes, um, 8 times. Wait, no, too big, so try 7; yeah, cool, that’s 154.”
All well and good, but mostly lies. First, that’s not “17,” it’s 1700. The question is NOT really “What is 22 into 17?” but rather, “how many groups of 1000 x 22 are there in 1730?” Oh, that surprises you? Well, why do you take for granted that you don’t have to ask that? The fact is that only experience, number sense or the mindless following of what teachers, parents, or peers told you lets you skip that question as unneeded. And similarly, probably without knowing it, you miss that the first question about 22 into 17 now becomes “How many groups of 100 x 22 are there in 1730?”
Again, the answer is 0, since 1 x 100 x 22 = 2200 which is too big. So finally, we should ask, “How many groups of 10 x 22 are there in 1730?” And the answer is 7. But please note well: that 7 represents “70,” which is 7 x 10, which is what we really multiply 22 by to get the first “partial product” of 1540 (see that I appended a zero to the 154 you calculated to make clear what is really being subtracted from 1730 on the first pass). Also, realize that using our method, that is the maximum we can justify subtracting, but we might have subtracted less! (See the discussion of partial quotients below).
So after we subtract 1540, we have not 19, but 190 remaining. The rest is pretty obvious (a dangerous word I’ll risk using here). The next step does not require lying: we really do ask,”How many groups of 1 x 22 are there in 190?” The answer is indeed 8. Any more would put us past 190. Our second partial product, from 8 x 22, is 176. We subtract that from 190 and are left with 14, too small for any more groups of 22 to be removed. That’s our remainder, then (alternatively, we can say that there are 14/22 of a group left, where a whole group has 22 in it).
So that is almost all we need to note save that what is missing from the process you were taught in school takes for granted one crucial idea: place value. We are first taking 70 groups of 22 from 1730 all at once, rather than doing it 70 times via individual subtractions of 22 at a time. And that’s a very good thing indeed. But because this process of compressing 70 subtractions into one estimation, one multiplication (or more if we estimate badly) and one subtraction hides information from us. “Better” algorithms are designed for efficiency, not for transparency. You snooze, you lose.
Except, we pretty much all snooze because it is almost unheard of for teachers to take the time to unpack this process, let alone teach it as repeated subtraction, then as partial quotients using place value, and then finally going to the more efficient but opaque standard algorithm. So for kids who don’t pick up the procedure, practice it to “mastery,” and then move on to the next disconnected procedure, long division is frequently a stumbling block. Even some students who can perform the algorithm correctly find it off-putting, I believe, given how negatively they react to the introduction of polynomial long division, a procedure that is, in practice, far easier to perform.
I would be remiss to conclude this essay without looking at one other approach, partial quotients: a blend of repeated subtraction and the standard long division algorithm. Let’s take the second example again, 1730 ÷ 22.
What if we weren’t sure about the exact procedure for the standard algorithm, but we felt that we knew the point of division (which is actually a lot more complicated and context-dependent than I have chosen to explore in this post): how many groups of 22 are there in 1730 with a leftover amount of from 0 to fewer than 22? (In general, we could call 22 the divisor, 1730 the dividend, 78 the quotient, and 14 the remainder, and in symbols, b = a*q + r, where b is the dividend, a is the divisor, q is the quotient, and r is the remainder, a, b, q, and r in Z, and 0 ≤ q < a).
So perhaps we do this:
What is THAT all about? Well, I thought, “Hmm, 10 x 22 = 220, so 20 x 22 = 440.” I then subtracted 1 x 20 x 22 and was able to repeat that twice more. But that left me with 410, too small for another subtraction of 20 x 22, but big enough to subtract 1 x 10 x 22. That left 190, too small for subtracting another 10 x 20 (predictably, given the previous analysis), but big enough to take away 1 x 5 x 22, or 110. That left 80, and I could see that 3 x 22 = 66 was small enough, while 4 x 22 = 88 would be too much to subtract. And with 14 left, I had to stop subtracting. Adding the partial quotients gave the complete quotient of 78 (groups of 22) plus the remainder, 14.
Before the forces of righteousness attack me for “dumbing down” America’s youth, I am not lobbying for “partial quotients” to replace the standard algorithm. Nor am I arguing against having students learn the standard algorithm, since, after all, calculators, computers, and smartphones do arithmetic, and much more, faster and more accurately. Instead, I am advocating for the end to black boxes when understanding is readily provided. How we get there, however, I will leave for another column.